Group Theory in the Bedroom

Having run out of sheep the other night, I found myself counting the
ways to flip a mattress. Earlier that day I had flipped the very
mattress on which I was not sleeping, and the chore had left a
residue of puzzled discontent. If you're going to bother at all with
such a fussbudget bit of housekeeping, it seems like you ought to do
it right, rotating the mattress to a different position each time,
so as to pound down the lumps and fill in the sags on all the
various surfaces. The trouble is, in the long interval between flips
I always forget which way I flipped it last time. Lying awake that
night, I was turning the problem over in my head, searching for a
golden rule of mattress flipping.

The essential characteristic of a golden rule is universality: One
rule works all the time, for everyone. The famous archetype of such
rules—the one about doing unto others—certainly has this
property. So does the rule of the road: "Drive to the
right." ("Drive to the left" works just as well; what
matters is not which side you choose but that everyone make the same
choice.) Not all rules generalize so smoothly. A sign saying
"Please use other door" is not helpful when it's posted on
every door. A golden rule of mattress flipping would be
some set of geometric maneuvers that you could perform in the same
way every time in order to cycle through all the configurations of
the mattress. Following this algorithm might entail more physical
labor on each occasion—perhaps making multiple flips or
turns—but at least it would eliminate the mental effort of remembering.

If you too spend sleepless nights fretting over this problem, I'm
afraid I have some disappointing news. You will not find the golden
rule of mattress flipping in this column. As a matter of fact, no
such rule exists—at least not in the form I originally
imagined. But please read on anyway: The search for a
mattress-flipping algorithm leads to some diverting mathematics, not
just in the bedroom but also in the garage and at the breakfast
table. Furthermore, although I can offer no golden rule for mattress
flipping, I do have some practical advice.